The concept of independent and dependent events comes into play when we are working on conditional probability. A compound or joint events is the key concept to focus in conditional probability formula. Drawing a card repeatedly from a deck of 52 cards with or without replacement is a classic example to explain these concepts.
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Independent and Dependent Events
Independent Events
Two independent events as disjoint sets; Ω denotes Sample Space.
An event that does not affect the occurrence of another subsequent event in a random experiment is an independent event.
- Ex. Tossing a coin.
- Here, Sample Space S = {H, T} and both H and T are independent events.
- Ex. Rolling a dice.
- Sample Space S = {1, 2, 3, 4, 5, 6}, all of these events are independent too.
Both of the above examples are simple events (events with single outcomes). Even compound events (two events occurring at the same time) can be independent events.
- Ex. Tossing a coin and rolling a die.
- Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.
- These events are independent because only one can occur at a time.
Dependent Events
When the occurrence of one event affects the occurrence of another subsequent event, the two events are dependent events. The concept of dependent events gives rise to the concept of conditional probability.
Browse more Topics Under Probability
- Independent and Dependent Events
- Mutually Exclusive Events
- Total and Compound Probability and Mathematical Expectation
Conditional Probability Formula
It is the probability of an event given that another event has already occurred. The event that occurred earlier may affect the probability of the present event under consideration. Also, the event happening later can occur only after the given former event has taken place.
How to denote?
If the probability of events A and B are P(A) and P(B) respectively then the conditional probability of B such that A has already occurred is P(A/B).
How to calculate?
P(A/B) = P(A ∩ B)/P(A) or P(B ∩ A)/P(A) given, P(A)>0
P(A)<0 means A is an impossible event. In P(A ∩ B) the intersection denotes a compound probability.
Illustration
- From a deck of 52 cards, a card is drawn randomly. We want to find the probability of getting a red card
- Let us denote this event as R1 and the probability as P(R1)
- Now if we again draw a card from the same deck what is the probability of getting a red card given that the card drawn earlier came out to be black and was not replaced into the deck
- The probability of getting a red this time is denoted as P(R2)
- conditional probability is P(R2/R1) = P(R1 ∩ R2) / P(R1)
- P(R1 ∩ R2) = P(R1) × P(R2)
- In case, replacement is done then P(R2/R1) = P(R2)
from the above information we can get the following relation:
- P(A ∩ B) = P(B/A) × P(A) ; P(A)>0
Solved Example on Conditional Probability Formula
Question. Out of 10 customers in a box 4 are purchased red bulbs, 3 purchased green bulbs and 2 purchased both green and red. If a customer at random bought a red bulb what is the probability that he/she also bought the green bulb?
Solution. The probability of consumers purchasing red from the question is P(R) = 4/10.
The probability of consumers purchasing both red and green from the question is P(R ∩ G) = 2/10.
So, from the given formula, the probability that a consumer buying red bulb is also buying the green bulb is
P(G/R) = P(G ∩ R)/P(R)
= 0.2/0.4
= 1/2 or 0.5
This concludes our discussion on the topic of conditional probability formula.
There are so many errors in two of the lectures that I have watched. The flow of the lectures are also inappropriate. Firstly you never defined what an event is. For this lecture you can just say that an event is a subset of sample space. Therefore it can be any subset of sample space, even phi(empty set) or the whole sample space itself. You are confusing events with elements of sample space.
There is a fundamental errors on tis page too. Like P(A|B) is probability of event A given that event B has (already) occurred. However the text in this page says the other way.
Please revise it before making public.
Your well-wisher
Kumar.
@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events.
Ex. Tossing a coin and rolling a die.
Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.”
The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample space is S as given but the even space is E= {(1,H),(1,T)} this is a compound event as the event has 2 possible outcomes that fit and not just one. ( and also they are independent.)
@Kumar Thank you! I thought I was somehow wrong in my understanding. Another error is: “Even compound events (two events occurring at the same time) can be independent events.
Ex. Tossing a coin and rolling a die.
Sample space S = {(1,H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.”
The example is still simple even and not a compound event. An actual example for a compound event will be ex: Tossing a coin and rolling a die and getting a 1 every time. The sample space is S as given but the even space is E= {(1,H),(1,T)} this is a compound event as the event has 2 possible outcomes that fit and not just one. ( and also they are independent.)